Previous work. From our knowledge there exist few studies on coordinate descent methods when the second term in the objective function is nonseparable. For example, [1]}, [2]}, [3]}, [4]} considers the composite optimization problem (REF ) with \(\psi \)  convex and separable (possibly nonsmooth) and the additional nonseparable constraints \(Ax=b\) . Hence, nonseparability comes from the linear constraints. In these settings, [1]}, [2]}, [3]}, [4]} proposed coordinate proximal gradient descent methods that require solving at each iteration a subproblem over a subspace generated by the matrix \(U \in \mathbb {R}^{n \times p}\)  using a part of the gradient of \(f\)  at the current feasible point \(x\) , \(\nabla f(x)\) , i.e:
\(\min _{d \in \mathbb {R}^p} f(x) + \langle U^T \nabla f(x),d \rangle + \frac{1}{2} d^T H_U d + \psi (x) \quad \text{s.t.} \quad AUd=0,\) 
